5 PROBABILITY SHORTCUTS FOR QUANT INTERVIEWS

1. For “at least one,” calculate none

Directly counting every way to get at least one success is messy. Use the complement: P(at least one) = 1 − P(none). The chance of at least one six in four fair rolls is 1 − (5/6)⁴ = 671/1296 ≈ 51.8%.

2. Add expectations, even when events interact

Linearity of expectation does not require independence. If 10 fair coins are flipped, the expected number of heads is simply 10 × 1/2 = 5. Define an indicator for each event, find each indicator's expected value, and add.

3. Look for symmetry before calculating

If two outcomes have identical roles, they often have identical probabilities. In a random ordering of two distinct cards A and B, A is before B exactly half the time. You do not need to enumerate the entire deck.

4. Rewrite “given that” as a smaller universe

Conditional probability means discarding outcomes that no longer fit. For two coin flips, given at least one head, the remaining equally likely outcomes are HH, HT, and TH. Only one is HH, so the probability of two heads is 1/3, not 1/2.

5. Count combinations when order does not matter

Ask whether rearranging the same selected objects creates a genuinely new result. Choosing 2 people from 8 gives C(8,2) = 8×7/2 = 28 pairs. Dividing by 2 removes the duplicate orders AB and BA.

A quick way to choose

• “At least one”: complement.
• “Expected number”: indicators and linearity.
• Equivalent players or positions: symmetry.
• “Given that”: shrink the sample space.
• Selecting a group: combinations.

Once the method feels obvious, try the probability section in our quant interview practice questions.